Proof of Proposition 7, p. 29 — Δ_H < Λ_{H1}

Let $(\theta_H,\theta_L,\alpha,\gamma,k,s)$ satisfy the standing assumptions $\theta_H>\theta_L>0$, $\alpha,\gamma\in(0,1)$, $k>0$, $s>0$. Then the high type's A-Learning threshold is strictly smaller than the first high-type O-Learning threshold:

This is the first of the three comparisons in the proof of Proposition 7: the high type's incentive to tell the truth binds at a lower ratio $\theta_H/\theta_L$ under A-Learning than under O-Learning when the seller-acceptance constraint binds.